Numerical data processing dedicated to an integrated microspectrometer

ABSTRACT

A method and apparatus for obtaining higher resolution spectral data on lower-resolution spectral data provided by a spectrometric transducer is described. The method includes calibrating the spectrometric transducer to produce results for a reconstruction of spectra using the results, and processing the lower-resolution spectra data using a set of numerical algorithms dedicated to an integrated micro-spectrometers associated with the spectrometric transducer and using the results of calibrating to provide the higher resolution spectra data. The apparatus utilizes the method.

This application claims the benefit of U.S. Provisional Application No.60/549,562, filed Mar. 4, 2004.

BACKGROUND OF THE INVENTION

A miniature low-cost integrated spectrophotometer (an integratedmicrospectrometer) may be used in a number of applications, such asindustrial monitoring and quality control, medical and pharmaceuticaltesting, plant growth characterization, environmental and pollutionmonitoring, food control, and light source testing. Process control,such as in chemical plants, in the semiconductor industry, in dye worksand in electroplating operations, is another area taking advantage ofhigher spectral resolution and measurement accuracy ofspectrophotometry. Other applications that may require simultaneousmeasurement of several spectra include multiangle color measurement ofpaint work, continual real-time monitoring of water quality, colorcontrol in color printers and digital cameras, olive-oil and winequality assessment, non-invasive blood glucose detection, fruitsanalysis, cosmetics and drug analysis, and security-related equipment.

Spectrophotometry may be considered as an analytic technique concernedwith the measurement and characterization of the interaction of lightenergy with matter. Spectrophotometry may involve working withinstruments designed for this purpose, referred to by some asspectrophotometers, and corresponding methods of interpreting theinteraction both at the fundamental level and for practical analysis.The distribution of light energy, absorbed or emitted by a sample of asubstance under study, may be referred to as its spectrum. If energy ofultraviolet (UV), visible (Vis) or infrared (IR) light is used, thecorresponding spectrum may be referred to as a light spectrum.

A spectrophotometer may have a resolution associated with its design orimplementation affecting resolution of measured spectra. As is wellunderstood by those of skill in the art of spectrometry, a requiredresolution for UV and a required resolution for IR spectral imaging maybe different. Further, the terms high-resolution and low-resolution arerelated to an imaged spectral band or to wavelengths of light within theimaged band. For a broadband spectrometer, either graduated spectralresolution or a spectral resolution sufficient to properly image eachband may be used.

Interpretation of spectra may provide fundamental information at atomicand molecular energy levels. For example, the distribution of specieswithin those levels, the nature of processes involving change from onelevel to another, molecular geometries, chemical bonding, andinteraction of molecules in solution may all be studied using spectruminformation. Comparisons of spectra to provide a basis for thedetermination of qualitative chemical composition and chemicalstructure, and quantitative chemical analysis is described in detail inParker S. (Ed.): McGraw-Hill Encyclopedia of Chemistry, McGraw-Hill,1983, which is hereby incorporated by reference.

Known techniques for processing data from spectrum analyzers tend toprovide inadequate quality of spectrum estimation for lower-costspectrophotometers.

U.S. Pat. No. 5,712,710 for example, describes a probe for use inmeasuring the concentration of a specific metal ion dissolved in liquid.The device suffers from known problems of probe miniaturization. Eitherthe bandwidth of the spectrometer is narrow to accommodate a small probesize, the quality of the spectral imaging is poor, or the opticalprocessing components are large and costly. The described devicecomprises a hand-held processing unit coupled to the probe. Theprocessing unit is programmed to calculate and display the concentrationof a specific material. In this probe, neither the photodetector nor theprocessing unit is integrated with the light diffraction structure.Further, the use of poor resolution in imaging the spectrum tends to beunacceptable for most applications when using such a probe.

U.S. Pat. No. 5,020,910 describes a method of forming a lightdiffraction structure directly over a photodetector. The describeddevice requires external electronic circuitry to obtain a usefulspectrum of light and the spectral resolution is very high in comparisonto that of existing conventional spectrometers.

U.S. Pat. No. 5,731,874 describes a spectrometer with an integratedphotodetector. The described device is sensitive only to particularspectral lines and thus tends to be useful only over a narrow spectralrange.

Therefore, there exists a need for a method and apparatus to alleviatesome disadvantages in the prior art.

Known laboratory spectrophotometers may perform acceptably for someapplications, but they are typically bulky and costly. It would bedesirable to provide lower-cost integrated micro-spectrophotometercapable of determining the spectral characteristics of the opticalsignals

SUMMARY OF THE INVENTION

In a broad aspect of the present invention, there is a family ofintegrated microspectrometers whose principle of functioning is based onthe subsequent use of a spectrometric transducer and a computingcircuit, including a digital signal processor.

In another broad aspect of the present invention, there is provided amethod for obtaining higher-resolution spectral data on thelower-resolution spectral data provided by the spectrometric transducer.The method comprises a set of numerical algorithms, dedicated tointegrated microspectrometers of this type, designed for calibration ofthose devices and for reconstruction of spectra using the results ofcalibration.

In another broad aspect of the present invention, there is provided amethod for obtaining higher resolution spectral data on lower-resolutionspectral data provided by a spectrometric transducer. The methodcomprises: calibrating the spectrometric transducer to produce resultsfor a reconstruction of spectra using the results; and processing thelower-resolution spectra data using a set of numerical algorithmsdedicated to an integrated micro-spectrometers associated with thespectrometric transducer and using the results of calibrating to providethe higher resolution spectra data.

BRIEF DESCRIPTION OF THE DRAWINGS

An embodiment of the present invention will now be described by way ofexample only with reference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram of an optical spectrum measurementinstrument.

FIG. 2 is a flow diagram of a procedure for resampling g_(n)(λ).

FIG. 3 is a flow diagram of a procedure for obtaining the final resultof a spectrum estimation.

DETAILED DESCRIPTION OF THE INVENTION

The description which follows, and the embodiments described therein,are provided by way of illustration of an example, or examples, ofparticular embodiments of the principles of the present invention. Theseexamples are provided for the purposes of explanation, and notlimitation, of those principles and of the invention. In thedescription, which follows, like parts are marked throughout thespecification and the drawings with the same respective referencenumerals.

A way to provide lower-cost integrated micro-spectrophotometer capableof determining the spectral characteristics of the optical signals isharness more of the computing power of microprocessors and otherprocessors. Aspects of such a design paradigm shift is described in U.S.Pat. No. 6,002,479 Apparatus and Method for Light Spectrum Measurementand U.S. Pat. No. 5,991,023 Method of Interpreting Spectrometric Data,the specifications of which are hereby incorporated by reference.Specific numerical techniques may be applied in utilizing thecomputational power of processors to ensure a required precision andhigh dynamic range of a spectrophotometer instrument, as describedbelow.

Although spectrometric transducers are used in a variety of devices, forthe purpose of the following discussion it will be described in thecontext of an integrated microspectrometer. As illustrated in FIG. 1, anintegrated microspectrometer 10 basically consists of a spectrometrictransducer 20 and a digital signal processor (DSP) 30. The spectrometrictransducer 20 converts an optical signal 12 into a digital signal yrepresentative of the spectrum x(λ) 15 of that optical signal 12. Thespectrometric transducer 20 receives an analog optical input 24 andprovides an output through N digital electrical outputs, typicallyphotodiodes that convert an incident optical signal into a correspondingelectrical signal. In practice, the spectrometric transducer 20 could bea dedicated optoelectronic transducer or a complete instrument such as aspectrum analyzer with fixed measurement parameters such as wavelengthrange, optical resolution, sensitivity, etc.

Assumptions and Notations

The spectrometric transducer 20 output-related discretization of thewavelength axis is defined by the sequence {λ_(n)} such that:λ_(min)=λ₁<λ₂< . . . <λ_(N−1)<λ_(N)=λ_(max)  Equation 1where N is the number outputs of the spectrometric transducer 20. Thus,the average interval between the consecutive wavelength values is:${\Delta\lambda} = \frac{\lambda_{\max} - \lambda_{\min}}{N - 1}$

The spectrometric transducer 20 calibration-related discretization ofthe wavelength axis is defined by the forrnula:λ_(m)′=λ_(min)′+(m−1)Δλ′ for m=1, . . . , M  Equation 2where M>>N is the number of discretization points, i.e. of the positionof the tunable monochromators used for calibration, and:${\Delta\lambda}^{\prime} = \frac{\lambda_{\max}^{\prime} - \lambda_{\min}^{\prime}}{M - 1}$with λ_(min)′≅λ_(min) and λ_(max)′≅λ_(max).

It is assumed that the spectrum x(λ) 15 of the analyzed optical signal12, i.e. light intensity vs. wavelength, may be adequately approximatedby a known function {circumflex over (x)}(λ; p) with unknown parametersp=[p₁ . . . p_(K)]^(T). A sequence of samples of x(λ) 15 is a particularcase of such a vector (due to the Shannon theorem). In this case:p≡x=[x ₁ . . . x _(K)]^(T) ≡[x(λ₁″) . . . x(λ_(K)″)]^(T)  Equation 3where:λ_(k)″=λ_(min)″+(k−1)Δλ″ for k=1, . . . Kwith λ_(min)″≅λ_(min) and λ_(max)″≅λ_(max), and:${\Delta\lambda}^{''} = {\frac{\lambda_{\max}^{''} - \lambda_{\min}^{''}}{K - 1}.}$Mathematical Model of the Data

An adequate implicit model of the data {{tilde over (y)}_(n)|n=1, . . ., N} 25 acquired at the output of the spectrometric transducer 20 hasthe form: $\begin{matrix}{{{F\left( {{\hat{y}}_{n};\alpha_{n}} \right)} = {\int_{{{- {\Delta\lambda}}/2}\rho}^{{{+ {\Delta\lambda}}/2}\rho}{\left\lbrack {\int_{- \infty}^{+ \infty}{g\left( {{\lambda_{n} + \lambda^{\prime}},\lambda} \right) \times (\lambda)\quad{\mathbb{d}\lambda}}} \right\rbrack\quad{\mathbb{d}\lambda^{\prime}}}}}{{{{for}\quad n} = 1},\ldots\quad,N}} & {{Equation}\quad 4}\end{matrix}$where F(●; α_(n)) is a known function with unknown parameters α_(n)=[α₁. . . α_(Q)]^(T), that models the inverse “static” characteristic of thespectrometric transducer 20 with respect to the nth photodiode (n=1, . .. , N); g(λ′, λ) is the impulse response ofthe optical part of thespectrometric transducer 20, and ρ is the ratio of the distance betweentwo consecutive photodiodes and the photodiode width. The above modelmay be given a simpler form: $\begin{matrix}{{{F\left( {{\hat{y}}_{n};\alpha_{n}} \right)} = {\int_{- \infty}^{+ \infty}{{g_{n}\left( {\lambda_{n} - \lambda} \right)} \times (\lambda)\quad{\mathbb{d}\lambda}}}}{{{{for}\quad n} = 1},\ldots\quad,N}} & {{Equation}\quad 5}\end{matrix}$by introducing the functions: $\begin{matrix}{{{g_{n}(\lambda)} = {\int_{{{- {\Delta\lambda}}/2}\rho}^{{{+ {\Delta\lambda}}/2}\rho}{{g\left( {{\lambda_{n} + \lambda^{\prime}},{\lambda_{n} - \lambda}} \right)}\quad{\mathbb{d}\lambda^{\prime}}}}}{{{{for}\quad n} = 1},\ldots\quad,N}} & {{Equation}\quad 6a}\end{matrix}$being the normalized responses of the photodiodes. Each of thesefunctions is the response of the spectrometric transducer 20, measuredat the output of the nth photodiode, to a sweeping monochromatorproducing an optical signal whose spectrum may be adequately modeledwith x(λ)≡δ(λ−1) where l is moving from λ_(min) to λ_(max). Thisresponse is assumed to be centered around λ=0, and—thus—satisfying thecondition: $\begin{matrix}{{{\int_{- \infty}^{0}{{g_{n}(\lambda)}{\mathbb{d}\lambda}}} = {{\int_{0}^{+ \infty}{{g_{n}(\lambda)}{\mathbb{d}\lambda}}} = 0.5}}\quad{{{{for}\quad n} = 1},\ldots\quad,N}} & {{Equation}\quad 6b}\end{matrix}$

Under this assumption, the response of the model to the optical signalwith a flat spectrum (i.e. constant as x varies), x(λ)=X^(f), satisfiesthe following equations:F(ŷ _(n); α_(n) ^(f))=X ^(f) for n=1, . . . , N  Equation 7and its response to a monochromatic signal, x(λ)=Xδ(λ−1), the equations:F(ŷ _(n); α_(n))=Xg _(n)(λ_(n)−1) for n=1, . . . , N  Equation 8Calibration of the Spectrometric Transducer

Numerical data processing dedicated to the integrated microspectrometer10 comprises reference data processing aimed at calibration of thespectrometrictransducer 20, and estimation of the spectrum of ananalyzed optical signal or of its parameters. Calibration-related dataprocessing may be performed by an external computer, while estimation ofparameters must rely on the internal DSP 30.

It follows from Equation 7 that the parameters α_(n) corresponding tothe nth wavelength (n=1, . . . , N) may be estimated during calibrationfrom the responses {{tilde over (y)}_(n,q) ^(f,cal)} of thespectrometric transducer 20 to Q flat-spectrum signals:x(λ)=x _(q) ^(f,cal)(λ)≡X _(q) ^(f) for q=1, . . . , Q  Equation 9by solving a set of algebraic equations: $\begin{matrix}{{{\hat{\alpha}}_{n} = {\arg_{\alpha}\left\{ {{{F\left( {{\overset{\sim}{y}}_{n,q}^{f,{cal}};\alpha} \right)} = {{X_{q}^{f}❘q} = 1}},\ldots\quad,Q} \right\}}}{{{{for}\quad n} = 1},\ldots\quad,N}} & {{Equation}\quad 10}\end{matrix}$Accordingly, α may be obtained for each n.

On the other hand, the estimation of the functions g_(n)(λ) could bebased on the reference data: $\begin{matrix}\left( {{{{\overset{\sim}{y}}_{n,m}^{cal}❘n} = 1},\ldots\quad,N} \right\} & {{Equation}\quad 11}\end{matrix}$acquired at the spectrometric transducer 20 output excited by abroadband source followed by a tunable monochromator:x _(m) ^(cal)(λ)=X _(m)δ(λ−λ_(m)′) for m=1, . . . , M  Equation 12

If the data are normalized before processing: $\begin{matrix}{\left. {\overset{\sim}{y}}_{n,m}^{cal}\Leftarrow{{F\left( {{\overset{\sim}{y}}_{n,m}^{cal};{\hat{\alpha}}_{n}} \right)}/X_{m}} \right.\quad{{{{for}\quad n} = 1},\ldots\quad,{N;\quad{m = 1}},\ldots\quad,M}} & {{Equation}\quad 13}\end{matrix}$then their model defined by Equation 8 takes on the form:$\begin{matrix}\begin{matrix}{{\hat{y}}_{n,m}^{cal} = {g_{n}\left( {\lambda_{n} - \lambda_{m}} \right)}} & {{{{for}\quad n} = 1},\ldots\quad,{N;{m = 1}},\ldots\quad,M}\end{matrix} & {{Equation}\quad 14}\end{matrix}$

This formula suggests that, for any fixed value of n, one may obtain adiscrete representation of g_(n)(λ) by direct smoothing of the data$\left\{ {{\left. {\overset{\sim}{y}}_{n,{M - m + 1}}^{cal} \middle| m \right. = 1},\ldots\quad,M} \right\}\text{:}$ĝ _(n,m) ≅g _(n)(λ_(n)−λ_(M−m+1)) for m=1, . . . , M  Equation 15

This sequence has to be properly centered by estimation of λ_(n). First,the interval └λ_(M−m*), λ_(M−m*+1)┘, the wavelength λ_(n) belongs to,should be identified:m*=arg _(m) {S _(n,m)≦0.5S _(n,m) _(max) ≦S _(n,m+1)}  Equation 16with: $\begin{matrix}{S_{n,m} = {\sum\limits_{\mu = m_{\min}}^{m}{\hat{g}}_{n,\mu}}} & {{Equation}\quad 17}\end{matrix}$where m_(min) and m_(max) are indices delimiting the central part of themain lobe of the estimate {ĝ_(n,m)} of the spectrometric transducer 20response corresponding to the n th wavelength value. Then, an estimate{circumflex over (λ)}_(n) of λ_(n) may be found by means of linearinterpolation based on two points:

λ_(M−m*) , S _(n,m*+1)

and

λ_(m−m*+1), S_(n,m*)

  Equation 18

The result of this interpolation is:centre of g {circumflex over (λ)} _(n) =w _(n)λ_(M−m*)+(1−w_(n))λ_(M−m*+1)  Equation 19with: $\begin{matrix}{w_{n} = \frac{S_{n,{m^{*} + 1}} - {0.5S_{n,m_{\max}}}}{{\hat{g}}_{n,{m^{*} + 1}}}} & {{Equation}\quad 20}\end{matrix}$which represents the centre of the response for a given output. As notedabove, the calibration α, to obtain α and g may be done on a computer.Estimation of the Spectrum and/or its Parameters

Using the results of the calibration, it is then possible to estimatespectrum using the DSP 30. It is assumed that the data for parameterestimation are normalized in the following way before processing:{tilde over (z)} _(n) =F({tilde over (y)} _(n); {circumflex over(α)}_(n)) for n=1, . . . , N  Equation 21

Consequently, the problem of spectrum estimation to be solved isnon-stationary but linear, and the parameterized model of the normalizeddata, resulting from of Equation 5, assumes the form:{circumflex over (z)} _(n)(p)=[g _(n)(λ)*{circumflex over (x)}(λ;p)]_(λ=λ) _(n) for n=1, . . . , N  Equation 22

A wide class of variational methods for spectrum estimation may bedefined using this model:

{circumflex over (p)}=arg_(p)inf{∥{tilde over (z)}(p)∥_(q)|pεP} with q=2or ∞  Equation 23where:{tilde over (z)}=[{tilde over (z)}₁ . . . {tilde over(z)}_(N)]^(T)  Equation 24{circumflex over (z)}(p)=[{circumflex over (z)} ₁(p) . . . {circumflexover (z)} _(N)(p)]^(T)  Equation 25and P is a set of optimization constraints.

Numerous sequential filtering algorithms may be directly applied oradapted for solving the problem of spectrum estimation on the basis ofraw measurement data modeled by Equation 22, in particular as describedin the attached bibliography:

-   -   direct methods [1], [2], [3], [4], [5];    -   spline-based recursive methods [6], [11];    -   Kalman-filter-based methods [7], [8], [9], [10], [11];    -   polynomial-filter-based methods [12], [13], [14], [15];    -   rational-filter-based methods [16], [17], [18];    -   Cauchy-filter-based and neural-network-based methods [19], [20],        [21].

However, under certain conditions, deconvolution methods may be alsoapplied for this purpose. Three viable options will now be described:transformation of the wavelength axis, interpolation of multiplesolutions and linear stationary solution.

Transformation of the Wavelength Axis

If the functions g_(n)(λ) differ in height to width ratio but do notdiffer significantly in shape, then deconvolution methods may be usedfor the estimation of the spectrum after appropriate transformation ofthe λ-axis:{overscore (λ)}=T(λ) for λε[λ_(min), λ_(max)]  Equation 26

This transformation should be based on the height-to-width ratioscharacterizing those functions: r₁, . . . , r_(N), and should convert anon-stationary problem of spectrum reconstruction into a stationary one;thus, it should satisfy the following conditions:T′(λ_(n))=ar _(n) +b for n=1, . . . , N  Equation 27T(λ₁)=ar ₁ +b  Equation 28T(λ_(N))=ar _(N) +b  Equation 29

Assuming that T(λ) is a known function, e.g. a spline function, with Nfree parameters, the above set of N+2 algebraic equations can be solvedwith respect to a, b, and those parameters. Consequently, the set oftransformed wavelength values can be determined:{overscore (λ)}_(n) T(λ_(n)) for n=1, . . . , N  Equation 30

They are necessary for resampling of the functions g_(n)(λ) andcomputing {{overscore (g)}_(m)}, which is the average of the resamplingresults. It should be noted that the dispersion in the resampledfunctions may be partially compensated for by appropriate adjustmentofthe function F({tilde over (y)}_(n); {circumflex over (α)}_(n)).

The procedure for resampling g_(n)(λ), represented by a sequence{g_(n,m)|m= . . . , −1, 0, 1, . . . }, is depicted by the flow chartshown in FIG. 2. The sequence of steps composing the procedure isindicated by the sequence of blocks 42 to 46.

In block 42 the procedure starts by determining {overscore(g)}_(n)({overscore (λ)}) by smoothing approximation or interpolation ofthe sequence {g_(n,m), {overscore (λ)}_(n+m)|m= . . . , −1, 0, 1, . . .}.

Then at block 44, {{overscore (g)}_(n,m)={overscore (g)}_(n)({overscore(λ)}_(n)+mΔλ)|m= . . . , −1, 0, 1, . . . } is computed.

Following which, at block 46,$\left\{ {{{\overset{\_}{g}}_{m} = {\left. {\frac{1}{N}\quad{\sum\limits_{n = 1}^{N}{\overset{\_}{g}}_{n,m}}} \middle| m \right. = \ldots}}\quad,{- 1},0,1,\ldots}\quad \right\}$is computed.

Assuming that the sequence {{overscore (x)}_(n)|n=1, . . . , N} is theresult of a deconvolution, obtained on the basis of {{tilde over(y)}_(n)} and {{overscore (g)}_(m)}, the procedure for obtaining thefinal result of the spectrum estimation is depicted by the flow chartshown in FIG. 3. The sequence of steps composing the procedure isindicated by the sequence of blocks 52 to 54.

In block 52 the procedure starts by determining {circumflex over(x)}({overscore (λ)}) by smoothing approximation or interpolation of thesequence {{overscore (x)}_(n), {overscore (λ)}_(n)|n=1, . . . , N}.

Then, at block 54, {{circumflex over (x)}_(n)={circumflex over(x)}(λ_(n))|n=1, . . . , N} is computed.

Interpolation of Multiple Solutions

An alternative methodology enabling the use of deconvolution algorithmsfor solving non-stationary problems of spectrum reconstruction may bebased on the use of different responses g_(i)(λ) for processing the datain consecutive overlapping intervals.

Assuming that:

-   -   each interval contains ΔN data;    -   the intervals are determined by the indices N_(i+1)=N_(i)+ΔN;    -   g_(i)(λ) is the response used for deconvolution for n=. . . ,        N_(i−1), . . . N_(i), . . . , N_(i+1), . . . ;    -   {{circumflex over (x)}_(n) ^((i))} is the result of        deconvolution performed with g_(i)(λ).

Then the linearly-interpolated final result of the deconvolution forn=N_(i−1), . . . N_(i), . . . , N_(i+1) may be calculated according tothe formula: $\begin{matrix}{{\hat{x}}_{n} = \left\{ {\begin{matrix}{{\frac{N_{i} - n}{\Delta\quad N}{\hat{x}}_{n}^{({i - 1})}} + {\frac{n - N_{i - 1}}{\Delta\quad N}{\hat{x}}_{n}^{(i)}}} & {{{{for}\quad n} = {N_{i - 1} + 1}},\ldots\quad,N_{i}} \\{{\frac{N_{i + 1} - n}{\Delta\quad N}{\hat{x}}_{n}^{(i)}} + {\frac{n - N_{i}}{\Delta\quad N}{\hat{x}}_{n}^{({i + 1})}}} & {{{{for}\quad n} = {N_{i} + 1}},\ldots\quad,N_{i + 1}}\end{matrix}.} \right.} & {{Equation}\quad 31}\end{matrix}$Linear Stationary Solution

The algorithms dedicated to spectrometric transducers ofrelatively highresolution may be developed taking into account that the imperfectionsof a monochromator and of a reference spectrometer used for calibrationcannot be neglected. On the other hand, those algorithms may be based onthe assumption that neither g_(n)(λ) nor α_(n), in Equation 5 depend onn. If, moreover, the assumption of the linear function F(●; α_(n)) isjustified, then the mathematical model of the spectrometric transducer20 and the mathematical model of the chosen reference spectrometer takeon the form:{circumflex over (z)}≡α ₀+α₁ ŷ _(n) =[g(λ)*x(λ)]_(λ=λ) _(n) for n=1, . .. , N  Equation 32{circumflex over (z)} _(R,n)≡α_(R,0)+α_(R,1) ŷ _(R,n) =[g_(R)(λ)*x(λ)]_(λ=λ) _(n) for n=1, . . . , N  Equation 33

By representing g(λ) as a convolution of g_(R)(λ) and an auxiliaryfunction g_(Δ)(λ):g(λ)=g _(R)(λ)*g _(Δ)(λ)  Equation 34the relationship between the two models becomes:{circumflex over (z)} _(n) =[g _(Δ)(λ)*z _(R)(λ)]_(λ=λ) _(n) for n=1, .. . , N  Equation 35where:z _(R)(λ)=g _(R)(λ)*x(λ).  Equation 36

Thus, the algorithm of spectrum reconstruction will be a numericalimplementation of the following deconvolution formula:{circumflex over (x)}(λ)=DECONV[{{circumflex over (α)} ₀+{circumflexover (α)}₁ {tilde over (y)} _(n) };ĝ _(Δ)(λ)]  Equation 37where: {circumflex over (α)}₀, {circumflex over (α)}₁, and ĝ_(Δ)(λ) arethe estimates of α₀, α₁, and g_(Δ)(λ), obtained during the calibrationof the spectrometric transducer 20.

The static calibration of the spectrometric transducer 20 requires theuse of two flat-spectrum signals:x ₁ ^(f,cal)(λ)≡X ₁ ^(f) and x ₂ ^(f,cal)(λ)≡X ₂ ^(f).  Equation 38

Assuming that X₂ ^(f)=0, the estimates of α₀, and α₁ may be obtained as:$\begin{matrix}\begin{matrix}{{\hat{\alpha}}_{1} = \frac{X_{1}^{f}}{{\overset{\_}{y}}_{1}^{f,{cal}} - {\overset{\_}{y}}_{2}^{f,{cal}}}} & {and} & {{\hat{\alpha}}_{0} = {{- {\hat{\alpha}}_{1}}{\overset{\_}{y}}_{2}^{f,{cal}}}}\end{matrix} & {{Equation}\quad 39} \\{{where}\text{:}} & \quad \\\begin{matrix}{{\overset{\_}{y}}_{1}^{f,{cal}} = {\frac{1}{N}\quad{\sum\limits_{n = 1}^{N}{\overset{\sim}{y}}_{n{.1}}^{f,{cal}}}}} & {and} & {{\overset{\_}{y}}_{2}^{f,{cal}} = {\frac{1}{N}\quad{\sum\limits_{n = 1}^{N}{{\overset{\sim}{y}}_{n,2}^{f,{cal}}.}}}}\end{matrix} & {{Equation}\quad 40}\end{matrix}$

The dynamic calibration of the spectrometric transducer 20 requires atleast one quasi-monochromatic signal x₁ ^(cal)(λ) and the acquisition ofthe corresponding reference data at the output of the spectrometrictransducer 20 and at the output of the reference spectrometer:$\begin{matrix}\left\{ {\overset{\sim}{y}}_{n,1}^{cal} \right\} & {and} & {\left\{ {\overset{\sim}{y}}_{R,n,1}^{cal} \right\}.}\end{matrix}$The estimate of g(λ) may then be determined by means of an algorithmbeing a numerical implementation of the following deconvolution formula:$\begin{matrix}{{\hat{g}(\lambda)} = {{DECONV}\left\lbrack {\left\{ {{\hat{\alpha}}_{0} + {{\hat{\alpha}}_{1}{\overset{\sim}{y}}_{n,1}^{cal}}} \right\},\left\{ {{\hat{\alpha}}_{R,0} + {{\hat{\alpha}}_{R,1}{\overset{\sim}{y}}_{R,n,1}^{cal}}} \right\}} \right\rbrack}} & {{Equation}\quad 41}\end{matrix}$where {circumflex over (α)}_(R,0) and {circumflex over (α)}_(R,1) are tobe determined in the same way as {circumflex over (α)}₀ and {circumflexover (α)}₁ on the basis of the responses of the reference spectrometerto two flat-spectrum signals.Global Optimization of the Microspectrometer Performance

Assuming that the spectrum x(λ) 15 is parameterized by equidistantsampling, as defined by Equation 3: $\begin{matrix}{{x(\lambda)} \cong {\sum\limits_{v = 1}^{N}{x_{v}\quad\sin\quad{{c\left( {\pi\quad\frac{\lambda - \lambda_{v}}{\Delta\quad\lambda}} \right)}.}}}} & {{Equation}\quad 42}\end{matrix}$

Consequently, the normalized model of the data defined by Equation 25takes on the form:{circumflex over (z)}=Gx  Equation 43where: $\begin{matrix}{{G_{n,k} = {\int_{- \infty}^{+ \infty}{{{g_{n}\left( {\lambda_{n} - \lambda^{\prime}} \right)} \cdot \sin}\quad{{c\left( {\pi\frac{\lambda^{\prime} - \lambda_{k}^{''}}{{\Delta\lambda}^{''}}} \right)} \cdot {\mathbb{d}\lambda^{\prime}}}}}}{{{{for}\quad n} = 1},\ldots\quad,{N;\quad{k = 1}},\ldots\quad,{K.}}} & {{Equation}\quad 44}\end{matrix}$

It follows from this model that the attainable accuracy of spectrumestimation will mainly depend onpon three factors:

-   -   the uncertainty Δ{tilde over (z)} of the vector {tilde over (z)}        whose elements are defined by Equation 21;    -   the uncertainty ΔG of the matrix G whose elements, defined by        Equation 44 are subject to the errors resulting from measurement        identification of the functions g_(n)(λ);    -   the conditioning number of the matrix G: cond(G)=∥G∥∥G⁻∥.

The following inequality characterizes this dependence: $\begin{matrix}{\frac{{\Delta\quad x}}{x} \leq {\frac{{{cond}(G)}\frac{{\Delta\quad G}}{G}}{1 - {{{cond}(G)}\frac{{\Delta\quad G}}{G}}} + {{{cond}(G)}\frac{{\Delta\quad\overset{\sim}{z}}}{\overset{\sim}{z}}}}} & {{Equation}\quad 45}\end{matrix}$provided ∥G⁻¹∥∥ΔG∥<1. (for any consistent pair of vector and matrixnorms).

The right-hand side of the inequality defined by Equation 45 should betaken into account as a criterion for global optimization ofspectrometric transducer 20 design.

INDUSTRIAL APPLICABILITY

It should be noted that the presented general methodology of numericaldata processing, dedicated to the integrated microspectrometer 10, takesinto account both random and systematic imperfections of thespectrometric transducer 20, including its nonlinearity and variabilityof its optical responses along the wavelength axis. This variability isof particular importance if the so-called multi-order effects in theoptical part of the spectrometric transducer 20 cannot be neglected.Their presence may be modeled by the multimodal functions g_(n)(λ), i.e.the functions having more than one maximum. The principal order isrepresented by a zero-centered peak, and higher and lower orders arerepresented by additional peaks whose positions are varying with thewavelength (n). The only change implied by this fact is the morecomplicated structure of the matrix G in Equation 43, sometimesresulting in worse illonditioning of the problem of spectrumreconstruction.

The following references address aspects of one or more embodimentsdescribed herein, and are hereby incorporated by reference:

-   -   [1] R. Z. Morawski, A. Podgórski: “Methodology of Investigation        of the Algorithms for Reconstruction of Thermokinetics”, Bull.        Pol. Acad. Sci.—Ser. Tech. Sci., 1983, Vol. 31, No. 1-12, pp.        65-69.    -   [2] R. Z. Morawski: “On reconstruction of Measurement Signals        with Discontinuities”, Proc. 5th Int. IMEKO-TC7 Symp. Intell.        Meas. (Jena, GDR, Jun. 10-14, 1986), Plenum Pub., 1987, pp.        257-260.    -   [3] R. Z. Morawski: “On Initial-condition Problem in Measurement        Signal Reconstruction”, Proc. 6th IMEKO-TC7 Int. Symp.        (Budapest, Hungary Jun. 10-12, 1987), pp. 47-52.    -   [4] R. Z. Morawski, A. Podgórski: “Results of Investigation of        Numerical Properties of an Algorithm for Reconstruction of        Thermokinetics”, Raporty Instytutu Radioelektroniki Politechniki        Warszawskiej, zeszyt 68, Warszawa 1984.    -   [5] R. Z. Morawski: “Metody odtwarzania sygnalów pomiarowych”,        Metrologia i Systemy Pomiarowe—Monografie, zeszyt 1, KMiAN PAN,        Warszawa 1989 (wyd. 1), 1990 (wyd. 2), stron 120.    -   [6] M. Ben Slima, R. Z. Morawski., A. Barwicz: “A Recursive        Spline-based Algorithm for Spectrophotometric Data. Correction”,        Rec. IEEE Instrum. & Meas. Technol. Conf.—IMTC'93 (Irvine, USA,        May 18-20, 1993), pp. 500÷503.    -   [7] D. Massicotte, R. Z. Morawski., A. Barwicz: “Efficiency of        Constraining the Set of Feasible Solutions in        Kalman-filter-based Algorithms of Spectrophotometric Data        Correction”, Rec. IEEE Instrum. & Meas. Technol. Conf.—IMTC'93        (Irvine, USA, May 18-20, 1993), pp.496-499.    -   [8] P. Brouard, R. Z. Morawski., A. Barwicz: “Approximation of        Spectrogrammes by Cubic Splines Using the Kalman Filter”, Proc.        1993 Canadian Conference on Electrical & Computer Engineering        (Vancouver, Canada, Sep. 14-17, 1993), pp. 900-903.    -   [9] P. Brouard, R. Z. Morawski, A. Barwicz: “DSP-based        Correction of Spectrograms Using Cubic Splines and Kalman        Filtering”, Record of IEEE Instrum. & Meas. Technol.        Conf.—IMTC'94 (Hamamatsu, Japan, May 10-12, 1994), pp.        1443÷1446.    -   [10] D. Massicotte, R. Z. Morawski, A. Barwicz.: “Incorporation        of a Positivity Constraint into a Kalman-filter-based Algorithm        for Correction of Spectrometric Data”, IEEE Trans. Instrum. &        Meas., February 1995, Vol. 44, No. 1, pp. 2-7.    -   [11] M Ben Slima, R. Z. Morawski, A. Barwicz:        “Kalman-filter-based Algorithms of Spectrophotometric Data        Correction—Part II: Use of Splines for Approximation of        Spectra”, IEEE Trans. Instrum. & Meas., June 1997, Vol. 46, No.        3, pp. 685-689.    -   [12] L. Szczeciński, R. Z. Morawski, A. Barwicz: “Spectrometric        Data Correction Using Recursive Quadratic Operator of Measurand        Reconstruction”, Proc. Int. Conf on Signal Processing        Applications & Technology—ICSPAT'95 (Boston, USA, Oct. 24-26,        1995), pp. 588-592.    -   [13] L. Szczeciński, R. Z. Morawski, A. Barwicz: “Quadratic FIR        Filter for Numerical Correction of Spectrometric Data”, Proc.        IEEE Instrum. & Meas. Technol. Conf—IMTC'96 (Brussels, Belgium,        Jun. 4-6, 1996), pp. 1046-1049.    -   [14] L. Szczeciński, R. Z. Morawski, A. Barwicz: “A Cubic        FIR-type Filter for Numerical Correction of Spectrometric Data”.        IEEE Trans. Instrum. & Meas., August 1997, Vol. 46, No. 4, pp.        922-928.    -   [15] L. Szczeciński, R. Z. Morawski, A. Barwicz: “Numerical        Correction of Spectrometric Data Using a Bilinear Operator of        Measurand Reconstruction”. Instrum. Sci. & Technol., 1997, Vol.        25, No. 3, pp. 197-205.    -   [16] L. Szczeciński, R. Z. Morawski, A. Barwicz: “Numerical        Correction of Spectrometric Data Using a Rational Filter”, J.        Chemometrics, Vol. 12, issue 6, 1998, pp. 379-395.    -   [17] M. Wiśniewski, R. Z. Morawski, A. Barwicz: “Using Rational        Filters for Digital Correction of a Spectrometric        Microtransducer”, IEEE Trans. Instrum. & Meas., Vol. 49, No. 1,        February 2000, pp. 43-48.    -   [18] M. P. Wiśniewski, R. Z. Morawski, A. Barwicz: “An Adaptive        Rational Filter for Interpretation of Spectrometric Data”, IEEE        Trans. Instrum. & Meas., 2003 (in press).    -   [19] P. Sprzqczak, R. Z. Morawski: “Calibration of a        Spectrometer Using a Genetic Algorithm”, IEEE Trans. Instrum. &        Meas., Vol. 49, No. 2, April 2000, pp. 449-454.    -   [20] P. Sprzqczak, R. Z. Morawski: “Cauchy-filter-based        Algorithms for Reconstruction of Absorption Spectra”, IEEE        Trans. Instrum. & Meas., Vol. 50, No. 5, October 2001, pp.        1123-1126.    -   [21] P. Sprzeczak, R. Z. Morawski: “Cauchy Filters versus Neural        Networks when Applied for Reconstruction of Absorption Spectra”,        IEEE Trans. Instrum. & Meas., Vol. 51, No. 4, August 2002, pp.        815-818.

Although the present invention has been described by way of a particularembodiment and example thereof, it should be noted that it will beapparent to persons skilled in the art that modifications may be appliedto the present particular embodiment without departing from the scope ofthe present invention.

1. A method for obtaining higher resolution spectral data onlower-resolution spectral data provided by a spectrometric transducer,comprising: calibrating the spectrometric transducer to produce resultsfor a reconstruction of spectra using the results; processing thelower-resolution spectra data using a set of numerical algorithmsdedicated to an integrated micro-spectrometers associated with thespectrometric transducer and using the results of calibrating to providethe higher resolution spectra data.